Problem: “I think I can make it. I'll just jump.” says Denis, flashing a grin as he looks to the neighboring rooftop. “Wait!” Vera exclaims. “First let's find the distance! Here, take a look at this diagram.” “I've never much cared for math,” Denis frowns. Vera rolls her eyes. “We know the distance between us, and we know the angles between ourselves and your target landing point. All we need to do is...” Denis jumps. “...apply the law of sines.” Vera gasps. According to Vera's diagram, how far did Denis need to jump to reach the neighboring rooftop? Do not round during your calculations. Round your final answer to the nearest tenth of a meter.
Converting the problem into geometrical terms Our problem can be modeled by the following triangle $\triangle ABC$, where we want to find $AC=d$. Because the interior angles of a triangle add to $180^\circ$, we know that $\angle C=35^\circ$. $105^\circ$ $\,40^\circ$ $35^\circ$ $4.7\,\text{m}\;\;$ $d$ $A$ $B$ $C$ Since we are given one side length and all angle measures, we can use the law of sines. Using the law of sines $\begin{aligned} \dfrac{\sin(C)}{AB}&=\dfrac{\sin(B)}{AC}\\\\ \dfrac{\sin(35^\circ)}{4.7} &= \dfrac{\sin(40^\circ)}{d} \gray{\text{Substitute}} \\\\ d \cdot \sin(35^\circ) &= 4.7 \cdot \sin(40^\circ) \\\\ d &= \dfrac{4.7 \cdot \sin(40^\circ) }{\sin(35^\circ)} \\\\ d &\approx 5.3 \,\text{m} \end{aligned}$ The answer Denis needed to jump $5.3 \,\text{m}$ to reach the neighboring rooftop.